I Space elevator and Coriolis force

[Prophet]

It seems to me that the concept of a space elevator does not take Coriolis force into account. If the elevator were in built with a space station in geosynchronous orbit and counterweight then there is more to reaching the space station than just climbing the rope. The rope would have to be anchored to the Earth at the equator and at the start of the climb would be moving approximately 1000 mph eastward. If memory serves geosynchronous orbit is about 25,000 miles from the center of the Earth and the space station would be moving eastward at over 6000 mph. I don't see how the elevator can gain that additional 5000 mph simply by climbing the rope.

I have read one article that acknowledges the Coriolis problem but they claim the effect is slight, simply pulling the rope slightly our of line. I don't think so.

I think you need to use Hamiltonian mechanics, not Newtonian mechanics, to solve but it's been over 40 years since I studied Hamiltonian mechanics.

Is NASA actually spending taxpayer money on this idea?

 

[stockzahn]

Supposing the space station is located directly above the Earth station of the elevator, the elevator only goes vertically upwards without changing its coordinates. The Coriolis acceleration in this case can be calculated with

$$a_C = 2\Omega v_{Elevator} cos(LAT)$$

where $\Omega = 2\pi/86400\,s$ denotes the rotational speed and $LAT$ denotes the latitude. The elevator is located close to the equator for obvious reasons, therefore $cos(LAT)\approx 1$. Now the Coriolis acceleration can be calculated depending on the speed of the elevator, e.g.

$v_{Elevator}=100\,m/s \rightarrow a_C = 0.015\,m/s^2$
$v_{Elevator}=1000\,m/s \rightarrow a_C = 0.15\,m/s^2$.

The accelerations obtained with this estimation seem to be manageable (if I didn't make a mistake).

 

[sophiecentaur]

I don't see how the elevator can gain that additional 5000 mph

The work needed to accelerate the vehicle would, I think, largely come from the spinning Earth (horizontal displacement of the tether) and not the fuel used for hoisting it.
At 100m/s, the journey would be a bit over 100hours or four days. (check my sums for 40 thousand km orbit) Not an exceptional time.

 

[DrStupid]

I have read one article that acknowledges the Coriolis problem but they claim the effect is slight, simply pulling the rope slightly our of line. I don't think so.

But that's exactly how it works. The coriolis force acting on the payload is not an issue. The challenge is fighting oscillations.

 

[DrStupid]

The work needed to accelerate the vehicle would, I think, largely come from the spinning Earth (horizontal displacement of the tether) and not the fuel used for hoisting it.

Above the geosynchronous orbit it actually comes from the spinning Earth only. And if the energy released by the payload above this point can be used to lift the payload below, than the elevator could theoretically work without external energy source or even be used as energy source itself.

 

[Khashishi]

As the payload goes up the cable, the cable will be providing the sideways force on the payload. But the reaction force on the cable will cause it to bow out. It will try to bounce back due to the tension in the cable which is provided by the Earth and the weight above geosynchronous orbit.
Without doing actual calculations, I suppose that raising an object will cause a wake in the cable that will propagate to the weight, which shifts the weight slightly west and down. But the cable will tense up and the weight will bounce back, causing the weight to oscillate. Eventually, the weight will pull the Earth along and lock up with the Earth's rotation. But since the Earth weighs much more than the weight, this could take a long, long time. Instead, some kind of active feedback will be needed on the weight, which will require considerable amounts of fuel.

 

[DrStupid]

Instead, some kind of active feedback will be needed on the weight, which will require considerable amounts of fuel.

There is no such feedback required if the cable is attached to the ground.